1. Field of the Invention
The present invention relates to a method of constraining stochastic models representing heterogeneous underground zones such as oil reservoirs to data referred to as dynamic because they vary with the fluid displacements. These data are, for example, production data or pressure data obtained from well tests.
2. Description of the Prior Art
The use of stochastic models of Gaussian type for representing the heterogeneity of underground structures is for example described by:    Journel, A. G. and Huijbregts, Ch. J.: “Mining Geostatistics”, Academic Press, 1978, or Chilés, J. P. and Delfiner, P.: “Geostatistics—Modeling Spatial Uncertainty”, Wiley-Interscience Publishers, John Wiley & Sons, 1999.
A numerical reservoir model can be formed from a set of grid cells to which the values of a realization of a stochastic model of Gaussian or related type are assigned. These values can be assimilated to porosities or permeabilities.
Matching the numerical reservoir model with the dynamic data measured in the field can be done as an optimization problem. A previously defined objective or cost function quantifies the difference between the dynamic data measured for the real medium and the corresponding responses of the numerical reservoir model. These responses are calculated by means of a numerical flow simulator. The goal of the optimization problem is to modify the reservoir model or rather the associated realization to minimize the objective function. This process is iterative: each iteration implies direct simulation of the flows. A good optimization method should allow: a) modifying realizations discretized on a very large number of grid cells; b) carrying out the modifications while respecting the stochastic model, that is the modified realization has to be coherent with the stochastic model; and c) limiting the number of direct flow simulations because they require a considerable calculating time.
Simulated annealing can be mentioned as an example of known optimization techniques. This approach is for example described by:    Gupta, A. D. et al.: “Detailed Characterization of Fractured Limestone Formation Using Stochastic Inverse Approaches”, SPE Ninth Symposium, 1994.
This technique is based on realization values exchange between grid cells. Upon each exchange, the objective function has to be calculated and therefore a direct flow simulation has to be carried out. This process requires an excessive number of iterations. Furthermore, in order to preserve the agreement between the realization and the stochastic model, an additional term concerning the variogram is introduced in the objective function, which makes optimization more delicate.
Other optimization techniques, more commonly applied, are based on calculation of gradients. Several approaches based on gradients are presented by: Tarantola, A.: “Inverse Problem Theory—Methods for Data Fitting and Model Parameter Estimation”, Elsevier Science Publishers, 1987.
They require calculating the gradients of the objective function with respect to the parameters of the problem which are the values of the realization at each grid cell. The realizations are then modified as a function of these gradients so that the objective function decreases. The problem related to conditioning of a reservoir model to production data is not linear: the minimization techniques using gradient calculation are used iteratively. After each modification of the realization, a direct flow simulation is carried out and the gradients are recalculated. Applied suddenly, the gradient methods lead to calibration of the dynamic data but they destroy the coherence between the stochastic model and the realization. Besides, the gradient methods do not allow consideration of a very large number of parameters. In order to overcome these limits, geostatistical parameterization techniques can be integrated thereto. The pilot point method can be mentioned at this stage, which is described by: de Marsily, G. et al.: “Interpretation of Interference Tests in a Well Field Using Geostatistical Techniques to Fit the Permeability Distribution in a Reservoir Model” in Verly, G. et al. (ed.), Geostatistics for Natural Resources characterization, Part 2, D. Reidel Pub. Co. 1984.
This method selects in the realization a certain number of points referred to as pilot points, in calculating the derivatives of the objective function with respect to the values at these points, in modifying the values of these points accordingly and in propagating the disturbance thus defined by means of a kriging technique. The pilot point method can induce deviant value variations of the pilot points.
Another geostatistical parameterization technique, which allows the above-mentioned difficulty to be overcome, is the method of gradual deformation of a stochastic model of a heterogeneous medium such as an underground zone. It is described and used by Hu, L.-Y. et al., in French patents 2,780,798 and 2,795,841 filed by the assignee.
The gradual deformation method allows gradual modification of a realization of a stochastic model of Gaussian or related type while respecting this model. The deformed realization still is a realization of the stochastic model. When the gradual deformation method is introduced in an optimization process, the procedure is as follows. The initial realization is combined with a fixed number of independent realizations related to the same stochastic model. These realizations are called complementary realizations. Combination is controlled by as many deformation parameters as there are complementary realizations. It produces a new realization. The derivatives of the objective function with respect to the deformation parameters are then calculated. The deformation parameters are modified so as to take into account the information from the derivatives. A first optimization in relation to the deformation parameters provides a realization verifying the stochastic model and reducing the objective function. In general, this optimization process has to be repeated several times with different complementary realizations so as to sufficiently reduce the objective function, which may require, in some cases, a prohibitive number of direct flow simulations.
The gradual deformation method allows, in some cases, modifying a realization locally. This possibility is justified when the gradual deformation is combined with the FFTMA geostatistical generator described by:    Le Ravalec, M. et al.: The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations, Math. Geol., 32(6), 2000.
This generator produces realizations for a stochastic model of Gaussian type specified beforehand by convoluting a Gaussian white noise with an operator depending on the covariance function. A local deformation can be carried out by applying the gradual deformation method to the Gaussian white noise underlying the realization.
The gradient techniques developed to date for calibration in relation to dynamic data are based on a direct link between the variations to which the realization representing the reservoir is subjected and the variation of the objective function. The optimization process involves modifying first the realization, then starting a flow simulation to apprehend the resulting variations for calibration.
A different approach valid in cases where the flows are modelled by streamlines, is proposed by:    Wang, Y. and Kovscek, A. R.: A Streamline Approach for History-Matching Production Data, SPE/DOE IOR, 2000.